Answer to Question #10310 in Algebra for nks prasad

2012-05-31T08:03:00-0400

Question #10310

using factor theorem, show that x-y, y-z, z-x are the factors of x^2(y-z)+y^2(z-x)+z^2(x-y)

Expert's answer

2012-06-05T08:10:34-0400

It is interesting property: if values of any two of three variables coincide then polynomial becomes 0.f(x,y,x)=f(x,x,z)=f(x,y,y)=0As we think about given f(x,y,z)= x^2(y-z)+y^2(z-x)+z^2(x-y) as polynomial with x - variable and y, z are parameters thenon value x=y we have f(y,y,z)=0, so (x-y)divides f;on value x=z we have f(z,y,z)=0, so (x-z)divides f;And if we consider f as polynomial from variable y and x,z are parameters then on value y=z we have f(x,z,z)=0, so (y-z)divides f;

We proved that (x-y),(x-z),(y-z) are factors of f(x,y,z) - given polynomial.